The Qur'an can claim whatever it wants. We also pray in Arabic in my church, so there is a lot of praying to Allah which of course no one has any problem with, since this is the word for God in Arabic. But Islam is something else entirely. It is within the self-interest of Muslims to present their god this way, of course, as their prophet did in trying to gather people to him and his religion by talking about "a common word between us (Muslims) and you (People of the Book), that we worship none but God" (from Surah al Imran). But he also said to the unbelievers (which includes in practice if not by definition the "People of the Book" [though this term has not itself had a consistent definition throughout Islamic history, as it has occasionally included people of religions other than Christianity and Judaism as far as was necessary for the establishment of Islamic government in certain places conquered by Muslims], since Islam maintains that we have somehow distorted our scriptures and their teachings to the point of being different today than we were in Muhammad's time when he praised certain Christians as being believers in God) that they do not worship what he worships and he does not worship what they worship (in Surah al Kafiroun, "The unbelievers"). I agree on this point, though obviously not in the way that Muslims intend it to mean...
You cannot allow yourself to become confused by the similar or same personages allegedly shared between this or that book. Of course Islam claims to be the religion of Abraham, Moses, Jesus, Mary, etc., because in Islamic prophetology everyone recognized by Islam/Muhammad as a previous messenger or pious religious figure before the actual establishment of Islam by Muhammad is therefore a Muslim or at least a proto-Muslim (what they call a hanif, in the case of certain Old Testament patriarchs; interestingly, this word ultimately comes from Syriac
hanapa, which meant in the original 'pagan; idolater"! See
Arthur Jeffery's classic "Foreign Vocabulary of the Qur'an" for many more such examples). So this is not surprising, and basing conclusions of who is worshiping what on such a surface-level understanding of how the religions and texts interact is foolhardy. We even have different names (in some cases, wildly so, from the point of view of Semitic word formation) for certain figures allegedly shared between Christianity and Islam when speaking in Arabic according to the religious identity of the speaker! So a Muslim will call the Islamic Jesus 'Isa, where the Christian calls the actual Jesus Yasou' (and Yasou' was known as a proper name prior to Islam, whereas 'Isa was not; Jeffery, mentioned earlier, connects the "Islamic" name to a likely origin in the East Syriac version of the name Jesus, 'Isha); a Muslim referring to John will call him "Yahya", but a Christian "Yuhanna", and so on. Some of the names are the same, though (Maryam, Ibrahim, Isma'il, etc). So you can't really base anything on what is shared by which religion based on who claims what personage for their own religion. It's not exactly
arbitrary (obviously Muslims very consciously do not have a St. Paul, as many claim he is responsible for corrupting their mythical 'original Christianity', which was essentially Islam before Islam; cf. above, regarding Islam's prophetology), but it doesn't actually tell us much about theological connections between any given religions.
Perhaps part of the confusion is that early commentators on Islam from a Christian perspective often saw it as a Christian-rooted heresy rather than its own separate religion (as John of Damascus, a contemporary to the first Muslim generation did in
his commentary on Islam). Some people do still see it that way today, though I don't think it's a majority view of any given church by any means. But still, the assumed common roots of the religion with Christianity and Judaism make sense within this view, as Arabia was never the sealed bubble of religious ignorance the Muslims like to portray it as in order to give their religion and its book and prophet some kind of excuse for existing. Indeed, there were many Arab Jewish and Arab Christian tribes, both before Muhammad and after him: the Ghassanids, eventually the Lakhmids (who would go on to establish the first Arab kingdom outside of the Arabian peninsula proper several centuries before Islam, at al Hira in what is now Iraq), the Banu Judham, and so forth.
But anyway...all of this is surface-level stuff, to explain why it's not enough to say "well this person in the Qur'an is the same as the one in the Bible, so they worship the same God." Not only do things not work that way, but it doesn't take much to see the Qur'an for what it really is, if you are versed in the history of the Bible. Consider for instance the Quran's reliance on apocryphal Christian literature popular in Muhammad's time and before, which is refashioned to support the Islamic narrative and presented as revelatory material from God, but is in fact transparently cribbed from these earlier sources. The most obvious example is probably the Arabic Infancy Gospel, a circa fifth-century text supposedly based on an earlier Syriac original, which contains the following passage:
"He has said that Jesus spoke, and, indeed, when He was lying in His cradle said to Mary His mother: I am Jesus, the Son of God, the Logos, whom thou hast brought forth, as the Angel Gabriel announced to thee; and my Father has sent me for the salvation of the world."
This scenario appears in a kind of distorted mirror image in the Qur'an (Surah 19:29-34), with Islam's Jesus figure, 'Isa, not coincidentally spouting an Islam-compliant message in contradiction to the Christian message of the earlier work:
"But she pointed to the babe. They said: "How can we talk to one who is a child in the cradle?" He said: "I am indeed a servant of Allah: He hath given me revelation and made me a prophet; And He hath made me blessed wheresoever I be, and hath enjoined on me Prayer and Charity as long as I live; (He) hath made me kind to my mother, and not overbearing or miserable; So peace is on me the day I was born, the day that I die, and the day that I shall be raised up to life (again)"! Such (was) Jesus the son of Mary: (it is) a statement of truth, about which they (vainly) dispute."
The Qur'an is a mishmash of various influences (some Christian, some Jewish, some pagan), tailored to fit its author's ideas about God and the proper worship of Him. To claim anything more than that, particularly linking the Islamic religion to other religions that do not accept it, says more about the people doing the linking than how congruent its message may be with the particulars of the revelation of God in other, earlier religions. Muhammad's supposed prophethood and therefore everything that has come out of it (i.e., Islam and the Qur'an) is not actually accepted by Jews or Christians of any era, so I'd say it's less a matter of being able to agree on the nature of God (which, from what I've understood, Jews and Muslims mostly do), and more a matter of saying that no new revelation, regardless of what it may incidentally get correct (y'know, a stopped clock being right twice and day and all that), is actually needed or accepted according to the people Islam tries to entice into its fold by taking earlier prophets and personalities and remolding them to fit the Islamic narrative.
